In the diagram, $AB$ is parallel to $DC,$ and $ACE$ is a straight line. What is the value of $x?$ [asy]
draw((0,0)--(-.5,5)--(8,5)--(6.5,0)--cycle);
draw((-.5,5)--(8.5,-10/7));
label("$A$",(-.5,5),W);
label("$B$",(8,5),E);
label("$C$",(6.5,0),S);
label("$D$",(0,0),SW);
label("$E$",(8.5,-10/7),S);
draw((2,0)--(3,0),Arrow);
draw((3,0)--(4,0),Arrow);
draw((2,5)--(3,5),Arrow);
label("$x^\circ$",(0.1,4));
draw((3,5)--(4,5),Arrow);
label("$115^\circ$",(0,0),NE);
label("$75^\circ$",(8,5),SW);
label("$105^\circ$",(6.5,0),E);
[/asy]
Explanation: Since $\angle ACE$ is a straight angle, $$\angle ACB=180^{\circ}-105^{\circ}=75^{\circ}.$$In $\triangle ABC,$ \begin{align*}
\angle BAC &= 180^{\circ}-\angle ABC - \angle ACB \\
&= 180^{\circ}-75^{\circ}-75^{\circ} \\
&= 30^{\circ}.
\end{align*}Since $AB$ is parallel to $DC,$ we have $$\angle ACD = \angle BAC = 30^{\circ}$$due to alternate angles. In $\triangle ADC,$ \begin{align*}
\angle DAC &= 180^{\circ}-\angle ADC - \angle ACD \\
&= 180^{\circ}-115^{\circ}-30^{\circ} \\
&= 35^{\circ}.
\end{align*}Thus, the value of $x$ is $\boxed{35}.$ [asy]
draw((0,0)--(-.5,5)--(8,5)--(6.5,0)--cycle);
draw((-.5,5)--(8.5,-10/7));
label("$A$",(-.5,5),W);
label("$B$",(8,5),E);
label("$C$",(6.5,0),S);
label("$D$",(0,0),SW);
label("$E$",(8.5,-10/7),S);
draw((2,0)--(3,0),Arrow);
draw((3,0)--(4,0),Arrow);
draw((2,5)--(3,5),Arrow);
label("$x^\circ$",(0.1,4));
draw((3,5)--(4,5),Arrow);
label("$115^\circ$",(0,0),NE);
label("$75^\circ$",(8,5),SW);
label("$105^\circ$",(6.5,0),E);
[/asy]